\section{Offline token-forwarding algorithms}
\label{sec:centralized}
We present two offline algorithms for $k$-gossip.  The first computes
an $O((n+k)\log^2 n)$-round schedule assuming that each node can send
at most one token to each neighbor in each round
(Section~\ref{sec:multiport}); the second computes an
$O(\min\{n\sqrt{k\log n}, nk\})$-round broadcast schedule assuming
that each node can broadcast at most one token to its neighbors in
each round (Section \ref{sec:upper}).  \junk{ and a bicriteria
  {$\rb{O(n^\epsilon), O(\log n)}$-approximation} algorithm in the
  broadcast model (Section \ref{sec:approx}).  } \junk{Both
  algorithms use a leveled graph constructed from the sequence of
  dynamic graphs which we call the {\em evolution graph}.
  Appendix~\ref{app:centralized} describes this construction and
  contains all omitted proofs.}

\junk{
}

\input{multiport_flow_based}
\input{flow_based}
%\input{approx}
